Killing vector fields of locally rotationally symmetric Bianchi type V spacetime

The classification of locally rotationally symmetric Bianchi type V spacetime based on its killing vector fields is presented in this paper using an algebraic method. In this approach, a Maple algorithm is employed to transform the Killing’s equations into a reduced evolutive form. Subsequently, the integration of the Killing’s equations is carried out subject to the constraints provided by the algorithm. The algorithm demonstrates that there exist fifteen distinct metrics that could potentially possess Killing vector fields. Upon solving the Killing equations for all fifteen metrics, it is observed that seven out of the fifteen metrics exclusively exhibit the minimum number of Killing vector fields. The remaining eight metrics admit a varying number of Killing vector fields, specifically 6, 7, or 10. The Kretschmann scalar has been computed for each of the obtained metrics, revealing that all of them possess a finite Kretschmann scalar and thus exhibit regular behavior.

of physical significance include Ricci collineations (RCs), which preserve the Ricci tensor; curvature collineations (CCs), which preserve the curvature tensor and matter collineations (MCs), which preserve the energymomentum tensor.These symmetries are characterized by the condition that the Lie derivative of the metric, Ricci tensor, energy-momentum tensor, and Riemann tensor along the symmetry vector field is zero 2 .In simpler terms, a KVF corresponds to a smooth vector field that preserve the metric of a spacetime metric.Mathematically η is said to be a KVF if, Furthermore, it is worth noting that a Lie derivative can also preserve a spacetime metric, although with the possibility of introducing a conformal factor multiplied by the metric itself.This type of symmetry is characterized by a conformal Killing vector field (CKVF).Mathematically, a vector field η is considered a CKVF if it fulfills the conformal Killing equation given by, where, α represents a real-valued function.The vector field η transforms into a KVF when α in Eq. (1.3) is equal to zero.Furthermore, when α in Eq. (1.3) is a constant, the vector field η defines a homothetic vector field.If we replace g pq with R pq in Eq. (1.3), it defines a conformal Ricci collineation, which reduces to an RC when α = 0 .If we consider the case when g pq is replaced by R pq and α is an arbitrary constant in Eq. (1.3), the vector field η transforms into a homothetic Ricci collineation.Similarly, by replacing g pq with T pq in Eq. (1.3), we can define a conformal matter collineation.The primary objective of this paper is to examine and discuss Killing vector fields.As a result, our discussion will be centered exclusively on KVFs.KVFs can be identified as solutions to Killing's Equation (1.2), which represents a system of 10 interconnected first-order partial differential equations.These equations are formulated in terms of four unknown functions that depend on four variables.When expressed in a coordinate system, the Killing equations take on a more convenient form, which can be stated as follows 2 , Within a 4-dimensional spacetime geometry, the maximum number of Killing vector fields is 10 when the spacetime metric is either flat or possesses constant curvature.Nevertheless, in the case of a non-flat spacetime geometry, it is anticipated that there will be less than or equal to seven Killing vector fields.
In order to emphasize the significance of Killing vector fields and their connections to other widely recognized symmetries, we provide a brief overview of recent literature.Bokhari et al. conducted a classification of spherically symmetric static spacetimes based on their Killing vector fields 3,4 .The authors, extended their research to explore the symmetry classification problem encompassing both RCs and CCs in the context of spherically symmetric spacetimes.Through their investigation, they established relationships between KVFs, RCs, and CCs 5,6 .In 7 , Moopanar and Maharaj explored conformal Killing vector fields of spherical spacetimes.A notable and extensive study conducted by Hussain et al. in 2019 focused on non-static spherically symmetric spacetimes and their conformal Ricci collineations 8 .
In 2003, Bokhari et al. extended the concept of Killing vector fields to provide a comprehensive classification of curvature collineations in cylindrically symmetric spacetimes 9,10 .Bokhari et al. extended their research on KVFs to explore matter collineations within a specific metric that exhibits static cylindrical symmetry 11 .In 2008, Bokhari et al. examined Killing symmetry of circularly symmetric static metric in three dimensions 12 .Feroze et al. carried out the classification of plane symmetric spacetimes by isometries 13 .Ziad presented the classification of static plane symmetric spacetime via their KVFs 14 .In 2004, Sharif investigated the symmetries of the energy-momentum tensor in static spacetimes with cylindrical symmetry 15 .In 2007, Shabbir et al. conducted a classification of static spacetimes with cylindrical symmetry based on their homothetic vector fields 16 .Ali and Feroze 17 achieved a comprehensive classification of static spacetimes with cylindrical symmetry based on conservation laws.
In this paper, we have classified LRS Bianchi type V spacetimes according to their Killing vector fields using the Rif tree approach.This approach utilizes the Rif algorithm, which is an algorithmic framework developed using the Exterior package in the Maple plate form.The Rif algorithm consists of a set of commands that facilitate the classification process.This algorithm presents a comprehensive set of conditions that encompass all possible constraints on the metric functions.The graphical representation is portrayed as a tree, known as a Rif tree, where each branch represents specific conditions on the metric functions that determine whether the spacetime can possess Killing vector fields.Subsequently, one must solve the set of Killing's equations, considering the conditions specified by the branches of the Rif tree, in order to obtain the explicit form of the Killing vector fields.Here we have obtained Killing algebras of dimension 4, 6, 7 and 10.

Classification of LRS Bianchi type V spacetime
The metric of LRS Bianchi type is given as 18 : This metric possesses the following four minimum KVFs: ,q + g rq η r ,p + g pq,r η r = 0 (2.1) The coefficients of the above metric are used in the definition of KVFs to obtain the set of Killing equations given below: In order to find the explicit form of KVFs, we need to solve these equations.Due to their non-linearity, these equations cannot be integrated directly without imposing some conditions on a(t) and b(t).For this purpose, we analyze these equations by Rif algorithm and as a result, we obtain the Rif tree given in Fig. 1.The expressions for nodes of the tree (pivots) are given in (2.12).
Using the conditions of each branch, we have solved Eqs.(2.2)-(2.11)that yield Killing algebras of dimensions 4, 6, 7 and 10.In branches 1, 2, 3, 4, 5, 6 and 14, we have obtained only four KVFs.In the forthcoming sections, we summarize the results of remaining branches based on the dimension of the obtained Killing algebras.

Six KVFs
This section contains metrics admitting six KVFs, four minimum and two additional ones.These metrics, along with their additional KVFs are presented in Tables 1 and 2.

Seven KVFs
In this section, we present some metrics with seven KVFs.These metrics, along with their additional three KVFs are given in Table 3.

Ten KVFs
There is only one metric, given by branch 13, that possesses ten KVFs.This metric is presented in Table 4 along with the solution of Killing equations and additional six KVFs.

Summary and discussion
In this paper, Killing vector fields have been calculated for LRS Bianchi type V spacetime.First we have used the metric of this spacetime in the definition of KVFs in order to obtain the set of ten Killing equations.A Maple algorithm, known as Rif algorithm was used to obtain a Rif tree along with the restrictions on the metric functions a(t) and b(t) under which the system of Killing equations has a solution.The algorithm gives fifteen different metrics which admit Killing vector fields of different dimensions.Solving Killing equations for all these metrics, we have observed that these metrics posses 4, 6, 7 or 10 dimensional Killing algebras.The Lie algebra for all the metrics possessing six KVFs is given by (2.12) where k 1 = 0 and www.nature.com/scientificreports/ Similarly, the Lie algebra for the metric admitting ten KVFs is obtained as where k 1 = 0 and where k 1 = 0 and where

Branch
Lie algebra of metrics with seven KVFs In order to add some physical implications, we find the source of matter for the obtained metrics and discuss their physical significance.It is well known that different forms of energy momentum tensor correspond to different matters.For example, T ab for a perfect fluid is of the form T ab = (ρ + p)u a u b + pg ab , where ρ , p and u a signifies energy density, pressure and the four velocity vector respectively.The four velocity vector u a for the LRS Bianchi type V metric gets the form u a = (1, 0, 0, 0) and hence T ab gives a perfect fluid if the non-diagonal component T 01 of energy-momentum tensor vanishes.Thus the metrics given by branches 8, 11, 12, 13 and 15 during our classification represent perfect fluid.For metrics of branches 8 and 13, the density and pressure become zero, giving vacuum solutions.The metric of branch 11 yields ρ = 3k 2 1 and p = −3k 2 1 .As ρ > 0, it shows that the metric is physically realistic.Moreover, the values of ρ and p satisfy all the energy conditions except the strong energy condition.For the metric of branch 12, the values of ρ and p are obtained as ρ = Here ρ > 0 provided that k 2 1 > m 2 and under this condition all the energy conditions are identically satisfied.Finally, for the metric obtained in branch 15, we have ρ = − 3m 2 k 2 , p = m 2 k 2 .As ρ < 0 , so this metric is an unrealistic metric.
In addition to the physical significance of the metrics discussed above, we also discuss the singularity of the obtained metrics.For this purpose, we calculate the Kretschmann scalar K = R abcd R abcd for all the obtained metrics, where R abcd is the Riemann tensor.A spacetime is said to be regular (non-singular) when its associated Kretschmann scalar possesses a finite value.Here, we check the singularity of the metrics which have more than four KVFs.
For the metric of branch 7, the obtained value of K is given by: Here the value of K is clearly finite and positive.So this metric has no singularity.The Kretschmann scalar K becomes zero for the metrics associated with branches 8 and 13, indicating that these two metrics exhibit regularity.
The metric associated with branch 9 yields the value of K as: Similar to the branch 7 metric, the metric of this branch is also regular, featuring a positive (finite) Kretschmann scalar.
The Kretschmann scalar K for the metrics corresponding to branches 10 and 11 is expressed as:  www.nature.com/scientificreports/It can be seen that the metrics of these branches are regular, having finite value of K.
In case of branch 12, the value of K is determined to be: which is evidently positive and finite, indicating a metric that is regular.The value of K for branch 15 gets the form: that demonstrates the absence of singularity in the metric.